Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts

Abstract

Abstract-In this paper we formulate tracking and stateestimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous trajectories arising from the impacts, we use hybrid systems stability analysis tools to establish the results. Using a novel concept of mirrored images of the target mass we prove that 1) a tracking control algorithm, and 2) an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards, including rectangles. Moreover, we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking. The results are illustrated via simulations.

Global Tracking and State Estimation with Nonsmooth Impacts for a Mass Confined to an N-Dimensional Half-Space

“…Given A cl Hurwitz, the results in Proposition 2 below are obtained for pairs (P, H) such that A T cl P + P A cl ≤ −H T H with (H, A cl ) observable (according to Property 1). The generality of allowing H T H ≥ 0 instead of requiring H T H > 0 is exploited (for the case n = 2) in Forni et al (2011a) to analyze some specific multiplewall billiards (parallel walls) for which the nonincreasing feature of the function W at jumps cannot be guaranteed by a matrix P which satisfies the stronger condition A…”

“…Goebel et al (2012) (see also Goebel et al (2009)) and it uses a novel concept of "mirrored reference", which is exploited in such a way that the tracking system decides what target (reference) should be tracked based on the minimum of several Lyapunov functions. Later, in Forni et al (2011a), this idea was further developed in such a way to exploit the recent results of Teel et al (2011) to assess exponential stability of the error dynamics of the tracking scheme, in addition to extending the results to the dual state estimation problem and combining the two schemes using a separation principle. Moreover, in Forni et al (2011a) the "min" rule introduced in Forni et al (2011b,c) was efficiently replaced by a suitable update law defined by an automaton.…”

“…Later, in Forni et al (2011a), this idea was further developed in such a way to exploit the recent results of Teel et al (2011) to assess exponential stability of the error dynamics of the tracking scheme, in addition to extending the results to the dual state estimation problem and combining the two schemes using a separation principle. Moreover, in Forni et al (2011a) the "min" rule introduced in Forni et al (2011b,c) was efficiently replaced by a suitable update law defined by an automaton. With the above extensions, the local asymptotic stability results of Forni et al (2011b,c) was extended in Forni et al (2011a) to global exponential stability.…”

“…Moreover, in Forni et al (2011a) the "min" rule introduced in Forni et al (2011b,c) was efficiently replaced by a suitable update law defined by an automaton. With the above extensions, the local asymptotic stability results of Forni et al (2011b,c) was extended in Forni et al (2011a) to global exponential stability.…”

Implicit numerical integration for the simulation and control of a non-smooth system with resets

“…However, the drawback is the low order of the method as against an event-driven scheme which can have a high-order in the free phase (when there is no impact). There has been some research efforts for studying these systems from a control perspective, including the design of controllers and observers [8], [9].…”